When pricing life annuity or insurance products issued to multiple lives, actuaries require a model for the survival of coupled lifetimes. For reasons of simplicity these multiple life premiums are often calculated under the assumption of independent lifetimes. In some circumstances this assumption is not realistic, and a number of correction methods based on bivariate survival data have been proposed in the actuarial literature. However, when we only observe the first occurring event time, dependent censoring occurs and the applicability of the models proposed for bivariate survival data turns out to be limited. The aim of this paper is to propose a copula-based analysis for bivariate survival data subject to left truncation and dependent censoring. Our method is based on parametric marginals and parametric copulas. For this model, we show that the association between the survival and dependent censoring time is identifiable from the distribution of the observed data. The proposed model is estimated using maximum likelihood, which simultaneously estimates the marginal and dependency parameters. We also develop a goodness-of-fit test approach to examine the validity of the fitted copula model. Finally, our approach is applied to a real data set on joint and last-survivor annuities from a large Canadian insurance company.